The bus at the end of the road says "World Tour of Camden". On the open-top deck is a band about to perform a song they first played 30 years ago. About 500 people in the street below are readying themselves to bounce up and down. It is late April. The sun is shining. Has there ever been a simpler recipe for happiness?

Madness take to the stairs at Dennis Severs' House. Photograph: Neil Wilder

Madness are alive again. A new album has reinvigorated an institution, and brought back a band member who said he had had enough. The live shows no longer suggest they are going through the motions, and the new material is some of the best they have ever written. And then there is something else - a track that is more than 10 minutes long, and the undeniable fact that Madness have made a concept album.

The concept is something they love - London - so where better to sing about it? They have prepared for the free street concert by spending the afternoon driving around in their bus, busking outside the Roundhouse and inside the Dublin Castle, the pub where they began playing in 1979. Steve Lamacq is broadcasting his BBC 6 Music show live from the bottom deck, the controller and head of music of Radio 2 and 6 Music are aboard, and Suggs is swigging from a flask as we drive along. "The early reviews are great," he says. "One of them calls it 'Madness's masterpiece'. I don't want to sound too pleased, but..."

But the reviews of the next album might be awful?

"Exactly! Enjoy it while we can!"

The new album, The Liberty of Norton Folgate, has enjoyed a long gestation. I had first heard about it late last year, on another journey through traffic in another form of London transport. "It is a concept album," Carl Smyth, the band's co-singer/songwriter, said in the back of a black cab. "I've been thinking for a while that Madness is the working-class Pink Floyd - therapy for the masses, a bright shining virus of joy."

Smyth, who is also known as Cathal Smyth and Chas Smash, and continues to anoint each live show with the call to flailing arms "Hey you! Don't watch that, watch this!", appears to be the band's main strategist; he is certainly its most eloquent philosopher. He says the new album combines Samuel Beckett, Bertolt Brecht, Ealing comedy, Tommy Cooper and Max Wall. He's been reading a lot of Peter Ackroyd and Iain Sinclair. "It's such a privilege to be in Madness," he says. "People have such love and affection for us."

Smyth is 50, and fond of quoting film-maker Akira Kurosawa's dictum that it is 40 years before a man knows his craft.

"I don't care about competing in that game, about record sales any more," he says. "Having worked in the industry [in management and A&R], I just see the people in it as jumped-up fucking furniture salesmen. For me, the true experience is live performance. I want us to be like the Grateful Dead, a massive nomadic touring entity which has a tribe around it: Madheads."

He says that when the band split up for four years in 1986, they felt they had become employees of the record company. They now have their own label, and a management that has engineered a good deal of creative freedom. "And we need good management: the band is seven people, but 21 fucking decisions. Madness is a slow behemoth when it moves. And it takes a long time to accept new people. But once we accept them it takes three times as long to get rid of them. People sense that we're a dysfunctional family, but we are a family. We're the good guys. The band have outlasted some of our marriages."

We are driving away from a photoshoot at the house of the artist Dennis Severs (1944-1999) in Folgate Street, a "still-life drama" that Severs reconstructed as if each dark and pungent room of the east London house were just as a Huguenot family of silk weavers had left it in the early 18th century. Smyth has had his photograph taken here before, a last family group shot to mark the end of his marriage. "I just thought it would be a nice memory," he says. "It was just over. I was crushed. I went into rehab to get my head together. I was so unhappy. I was 16 stone. I thought if I hang around I'll just start doing drugs and drink. Rehab cost me a fortune, but it was the best thing I ever did. I found out the core issues: why I behave the way I do. Changed my life. Lost weight. Got focused." Smyth also got a place in Ibiza. "I thought, 'I'm not paying money starting again, getting a mortgage together to live in a rabbit hutch.' The kids love Ibiza [he has three]. Got happy with myself again. Quality of life - that's what it's all about for me now."

We are heading towards the Hackney Empire, where Madness are unveiling their new songs in a theatrical show; the first half is The Liberty of Norton Folgate - all Dickensian vaudeville with dark shadows and choreographed street performers (Julien Temple is around to film it); the second half is House of Fun, It Must Be Love, Night Boat to Cairo. Smyth hands me a programme he has composed for the show - a collection of free-form stories glowing with cobbled nostalgia ("Today the East India Dock Co has a most significant arrival from the Far East... silks and jade... and it has been whispered an ointment made from the eyelashes of a dragon!").

"I think it's about time that people realised that Madness is a lot smarter than they think," Smyth says as he begins an explanation of the album's title. But then his phone rings, a friend wanting tickets. "Tonight is smashed, mate," Smyth tells him. "It's all celebs. Paul and Mick are coming [Simonon and Jones]. I'll try you for Thursday. Yes, that's tomorrow, darling. Ring me tomorrow first thing... well, not first thing as I'll be asleep. About two or three. Frank, do me a favour will you? Fuck off! Talk to you tomorrow, OK?"

He puts the phone back in his jacket. "It always gets mental towards showtime, and it's always people you haven't spoken to for years. I've got fans who have my number and they text me when they have a baby. We've never played that game of 'when we arrive back at the airport send the fan club down'. I used to reply to fans on the website but I had to close it down because you just become an agony aunt. But it goes with the territory - it's not bad. My grandad was down the pit and got coal lung, so relative to that this is a doddle."

Smyth first read about the Liberty of Norton Folgate in a novel by Patrick O'Brian. An area of about nine acres in Spitalfields and Shoreditch, it once housed its own governing council and judiciary, and offered a refuge to vagabonds and the creative who considered themselves outside or beyond the law. It was abolished in 1900, but there are plans to reconvene the government for the launch of the album. Smyth is dismayed by what has become of the area -"all these greedy developers, all this glass uglinesss" - and he hopes his band's efforts may reclaim some former glories. The title song's lyrics recall the delights and dangers of music hall, boxing ring and moody street market, and the wash of humanity drifting up from the Thames. In other hands it could be banal; in the hands of Madness it sounds magnificent.

"I do feel this is our coming out record," Suggs says at the back of the Hackney Empire. "This is where we're meant to be. We've tried to make it an amalgam of all the things we've loved, and thought, 'How can we stretch it a bit?' We've hardly ever done a song over three minutes, as we've always hated long songs with a vengeance. But Norton Folgate is really a deceit - it's four songs stuck together. At least there aren't any guitar solos that go on for weeks."

He's a little hungover this afternoon. A blow-out the night before had the band talking about taking the Norton Folgate show to regular theatres - a fortnight in Paris, a week in Berlin - the optimism fuelled by the first show the night before. "It could have been a complete disaster. If after four minutes people started shouting 'Baggy Trousers!' then the balloon would have popped and it was all over. The whole thing would have just collapsed in self-conscious hatred of each other. Going into that void and being absolutely shit scared, but then it was this great feeling..."

Suggs is 48. He has branched out into radio and TV presenting, fearing the singing career might be over by middle age. He remembers seeing the Rolling Stones in their 40s and thinking, "miserable old cunts", but he has since found other inspirations. "My epiphany was seeing Buena Vista Social Club, with that guy in his 80s who's dead now (Ibrahim Ferrer), and he came on stage with two walking sticks. But the music started, he threw the sticks in the air and went into one. I thought, 'Well, there is a dignified way of doing this. You don't have to be a juvenile buffoon.'"

Chris Foreman, the group's guitar player, found no such solace. A few years ago he quit the band with complaints of "the petty, time-consuming bollocks" that drained their creativity. "I left because we were doing this album of ska covers (as the Dangermen), and, to me, it seemed a bit desperate. We started off doing really quirky stuff, and then one day we're back doing Israelites. And a couple of the guys were getting on my nerves a bit. I said to them not to put 'Going for a sabbatical' in any press releases, but of course they did. And I suppose it was in a way..."

He enjoyed his new freedoms - he moved to Brighton, and when he picked the milk from his doorstep it made him feel like Henry Hill in Goodfellas. "I had a new identity. I didn't really need the money, and I can just sit looking out the window all day. Some people in the band are very driven, and I'm not." But he returned in 2006, drawn by the strength of the new material.

"Because just doing the old stuff all the time is a bit of a doddle,' he says. "In Grey Day, where I have a guitar solo, sometimes I start that and I can't be bothered really. Sometimes I start it and I know it's going to be crap. And there is no way I can stop it being crap."

Mike Barson, the keyboard player and co-songwriter, also left the band for a while and moved to Holland. "All the musical ideas just dried up," he remembers. "We became so obsessed about doing something different that we ended up down a dead-end. We got to a point where we decided never to repeat ourselves, but the room just got smaller and smaller."

Barson now spends some of his life in France, on a Buddhist retreat, where no one knows who he is. "Buddhism makes a lot of sense out of life, I think, an intelligent spirituality which doesn't clash with living a normal day-to-day life. But I love the music more than ever now. We get a fair amount of respect these days, and in the beginning perhaps we didn't. Recently we were rehearsing and Ray Davies was next door, and at the end we were talking, and I think he respected us. I hope I haven't got that wrong, Ray."

Barson is speaking backstage before another show, and he too is thrilled with his band's reinvention. "Finally I think we've got it right,' he says. "It takes a bit of thought about how to present new material. When our second album Absolutely was about to come out, and we said, 'Here's a new one,' the audience just sighed and went, 'Oh no, bloody hell.' It went down like a lead balloon. And that was Baggy Trousers."

At the Camden Crawl a few weeks ago Baggy Trousers sounded rather good. And the new material sounded like its equal. There was something quaint about seeing fully-grown men and women dancing to songs about schooldays as if they were still in the midst of them. And it was touching too: the audience, like the band on the bus in front of them, had survived, and who was to say that the best was not yet to come?

Madness: a nutty life

1979 The North London Invaders change their name to Madness. Between 1979 and 1983, they land 14 top 10 singles, only one of which, House of Fun, reaches No 1, in May 1982.

June 1984 The band's founder and main songwriter Mike Barson leaves the band.

September 1986 Madness split, citing "musical differences".

May 1988 Four members - Suggs, Lee Thompson, Chas Smash and Chris Foreman - reappear, briefly, as The Madness.

August 1992 A reunion show, Madstock!, takes place in Finsbury Park, north London. Three more Madstocks! follow over the next six years.

October 2002-August 2003 Our House, a musical based on their songs, runs at London's Cambridge Theatre.

2004 Play a series of low-key shows as the Dangermen, covering vintage ska songs.

May 2009 Release The Liberty of Norton Folgate, their first album in 10 years.